Theorem xpss1 4857

Description: Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)

Assertion

Ref	Expression
xpss1	⊢ (A ⊆ B → (A × C) ⊆ (B × C))

Proof of Theorem xpss1

Step	Hyp	Ref	Expression
1		ssid 3291	. 2 ⊢ C ⊆ C
2		xpss12 4856	. 2 ⊢ ((A ⊆ B ∧ C ⊆ C) → (A × C) ⊆ (B × C))
3	1, 2	mpan2 652	1 ⊢ (A ⊆ B → (A × C) ⊆ (B × C))

Syntax hints:   → wi 4   ⊆ wss 3258   × cxp 4771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-opab 4624  df-xp 4785

This theorem is referenced by:  ssres2  4992  funssxp  5234